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On 14/08/2012 02:51, Mike Dougherty wrote:
<blockquote
cite="mid:CAOJFdbLC+veoCBtQNcMwEj+0ARvXqAK7XiuwnD6CrDSALOoqNw@mail.gmail.com"
type="cite">
Is there a relationship to fractal imagery? Why are the numbers
prime?</blockquote>
<br>
There is a theorem by Henderson that states that a necessary
condition for the existence of a symmetric n-Venn diagram is that n
is a prime number.<br>
<br>
<span style="color: rgb(0, 0, 0); font-family: 'Times New Roman';
font-size: medium; font-style: normal; font-variant: normal;
font-weight: normal; letter-spacing: normal; line-height: normal;
orphans: 2; text-align: left; text-indent: 0px; text-transform:
none; white-space: normal; widows: 2; word-spacing: 0px;
-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;
background-color: rgb(255, 255, 255); display: inline !important;
float: none; ">D. W. Henderson,<span class="Apple-converted-space"> </span></span><em
style="color: rgb(0, 0, 0); font-family: 'Times New Roman';
font-size: medium; font-variant: normal; font-weight: normal;
letter-spacing: normal; line-height: normal; orphans: 2;
text-align: left; text-indent: 0px; text-transform: none;
white-space: normal; widows: 2; word-spacing: 0px;
-webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;
background-color: rgb(255, 255, 255); ">Venn diagrams for more
than four classes</em><span style="color: rgb(0, 0, 0);
font-family: 'Times New Roman'; font-size: medium; font-style:
normal; font-variant: normal; font-weight: normal; letter-spacing:
normal; line-height: normal; orphans: 2; text-align: left;
text-indent: 0px; text-transform: none; white-space: normal;
widows: 2; word-spacing: 0px; -webkit-text-size-adjust: auto;
-webkit-text-stroke-width: 0px; background-color: rgb(255, 255,
255); display: inline !important; float: none; ">, American
Mathematical Monthly, 70 (1963) 424-426.</span>
<a class="moz-txt-link-freetext" href="http://www.jstor.org/stable/2311865">http://www.jstor.org/stable/2311865</a><br>
<br>
This later paper (which is freely accessible) states the basic
argument (which is very clean and simple), and then goes on patching
a serious hole in it:<br>
<a
href="http://www.math.umn.edu/%7Ewebb/Publications/monthly645-648-wagon.pdf">http://www.math.umn.edu/~webb/Publications/monthly645-648-wagon.pdf</a><br>
<br>
<br>
Fractals: I think this is mainly because of how the human mind
works. We are bad at maintaining large amounts of information in
working memory, so we like to work with modular or recursive
objects: while the object is complicated, the method of making it is
simple. So the successful methods of making multidomain Venn
diagrams (see the examples on Wikipedia, <a
href="http://en.wikipedia.org/wiki/Venn_diagram">http://en.wikipedia.org/wiki/Venn_diagram</a>
) all tend to produce patterns that are fractal-like. <br>
<br>
In general, I expect that fractals often emerge because the universe
also tends to follow simple rather than complex programs. Even when
these programs make organisms it is easier to grow and evolve
recursive structures than non-recursive ones. So the important and
deep question is what forces affect what programs get run, and what
kind of simplicity is optimized for. <br>
<br>
<br>
BTW, here is a stained glass Venn diagram from Gonville and Caius
college in Cambridge:<br>
<a href="http://www.flickr.com/photos/arenamontanus/3212655985/">http://www.flickr.com/photos/arenamontanus/3212655985/</a><br>
I was eating breakfast in the hall when I noticed a familiar diagram
in the stained glass windows. Then I noticed the other windows:<br>
<a
href="http://www.flickr.com/photos/arenamontanus/3213502484/in/photostream/">http://www.flickr.com/photos/arenamontanus/3213502484/in/photostream/</a><br>
<a
href="http://www.flickr.com/photos/arenamontanus/3213502248/in/photostream/">http://www.flickr.com/photos/arenamontanus/3213502248/in/photostream/</a><br>
Venn, like the others, was a fellow there. <br>
<br>
<pre class="moz-signature" cols="72">--
Anders Sandberg,
Future of Humanity Institute
Philosophy Faculty of Oxford University </pre>
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